# Find the second partials (including the mixed partials) f(u,v)=18In(u^2+v^2)fuu=_____,fvv=______,fuv=fvu=_______

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### 2 Answers

You need to evaluate the first order partial derivatives, such that:

`f_u = (del(18ln(u^2+v^2)))/(del u)`

`f_v = (del(18ln(u^2+v^2)))/(del v)`

You need to evaluate `f_u` differentiating the given function with respect to u, using chain rule and considering v as constant, such that:

`f_u = 18/(u^2+v^2)*(del(u^2+v^2))/(del u)`

`f_u = (36u)/(u^2+v^2)`

You need to evaluate `f_v` differentiating the given function with respect to v, using chain rule and considering u as constant, such that:

`f_v = 18/(u^2+v^2)*(del(u^2+v^2))/(del v)`

`f_v = (36v)/(u^2+v^2)`

You need to evaluate the second order derivative `f_(u u) ` differentiating f_u with respect to u, using quotient rule and considering v as constant, such that:

`f_(u u) = ((36u)'(u^2+v^2) - (36u)(u^2+v^2)')/(u^2+v^2)^2`

`f_(u u) = (36u^2 + 36v^2 - 72u^2)/(u^2+v^2)^2`

`f_(u u) = (36v^2 - 36u^2)/(u^2+v^2)^2`

You need to evaluate the second order derivative `f_(v v)` differentiating `f_v` with respect to v, using quotient rule and considering u as constant, such that:

`f_(v v) = ((36u)'(u^2+v^2) - (36u)(u^2+v^2)')/(u^2+v^2)^2`

`f_(v v) = (36u^2 + 36v^2 - 72v^2)/(u^2+v^2)^2`

`f_(v v)= (36u^2 - 36v^2)/(u^2+v^2)^2 = -f_(u u)`

You need to evaluate the second order derivative `f_(u v)` differentiating `f_u` with respect to v, using quotient rule and considering u as constant, such that:

`f_(u v)= ((36u)'(u^2+v^2) - (36u)(u^2+v^2)')/(u^2+v^2)^2`

`f_(u v)= (0 - (36u)(2v))/(u^2+v^2)^2`

`f_(u v)= (-72uv)/(u^2+v^2)^2 = f_(v u)`

**Hence, evaluating the second order partial derivatives yields **`f_(v v)= -f_(u u) = (36u^2 - 36v^2)/(u^2+v^2)^2; f_(u v)= f_(v u) = (-72uv)/(u^2+v^2)^2`

`f(u,v)=18ln(u^2+v^2)`

differentiate f w.r.t. u anv partiallly

`f_u=(18/(u^2+v^2))(u^2)'`

`f_u=(18/(u^2+v^2))(2u)`

`f_u=36u/(u^2+v^2)`

`f_(u u)=36{(u^2+v^2)-u(2u)}/(u^2+v^2)^2`

`` `f_(u u)=36{(u^2+v^2-2u^2)/(u^2+v^2)^2}`

`f_(u u)=36{(v^2-u^2)/(u^2+v^2)^2}`

`f_v=(36v)/(u^2+v^2)`

`f_(v v)=(36((u^2+v^2)-v(2v)))/(u^2+v^2)^2`

`f_(v v)=(36(u^2-v^2))/(u^2+v^2)^2`

`f_u=36u(u^2+v^2)^(-1)`

`f_(uv)=36u(-1)(u^2+v^2)^(-2)(2v)`

`f_(uv)=-72uv(u^2+v^2)^(-2)`

`f_v=36v(u^2+v^2)^(-1)`

`f_(vu)=36v(-1)(u^2+v^2)^(-2)(2u)`

`f_(vu)=-72vu(u^2+v^2)^(-2)`

`f_(u u)=36(v^2-u^2)/(u^2+v^2)^2,f_(v v)=36(u^2-v^2)/(u^2+v^2)^2`

`f_(uv)=f_(vu)=-72(uv)/(u^2+v^2)^2`

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